Strengthening Hadwiger's conjecture for 4- and 5-chromatic graphs

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY Accounts of Chemical Research Pub Date : 2023-09-12 DOI:10.1016/j.jctb.2023.08.009

Anders Martinsson, Raphael Steiner

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Abstract

Hadwiger's famous coloring conjecture states that every t-chromatic graph contains a $K_{t}$ -minor. Holroyd [11] conjectured the following strengthening of Hadwiger's conjecture: If G is a t-chromatic graph and $S \subseteq V (G)$ takes all colors in every t-coloring of G, then G contains a $K_{t}$ -minor rooted at S.

We prove this conjecture in the first open case of $t = 4$ . Notably, our result also directly implies a stronger version of Hadwiger's conjecture for 5-chromatic graphs as follows:

Every 5-chromatic graph contains a $K_{5}$ -minor with a singleton branch-set. In fact, in a 5-vertex-critical graph we may specify the singleton branch-set to be any vertex of the graph.

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加强4-色图和5-色图的Hadwiger猜想

Hadwiger著名的着色猜想指出，每个t-色图都包含一个Kt小调。Holroyd[11]猜想了Hadwiger猜想的以下加强：如果G是一个t-色图，并且s⊆V（G）在G的每个t-色中取所有颜色，那么G包含一个根在s的Kt小调。我们在t=4的第一个开放情况下证明了这一猜想。值得注意的是，我们的结果还直接暗示了Hadwiger关于5-色图的猜想的一个更强版本如下：每个5-色图都包含一个带有单例分支集的K5小调。事实上，在5顶点临界图中，我们可以将单例分支集指定为图的任何顶点。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.

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